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Unformatted text preview: Math 602 Homework 7 The problems here are well known results. You need to show all your work and explanations. 1. (i) Check that the functions 1, sin(nx), cos(nx), (n = 1, 2, 3 . . .) form an orthogonal system in L2 [-, ]. (ii) Normalize them to obtain an orthonormal system. (iii) Assuming that f L2 [-, ] has its Fourier series expansion f= a0 + [an cos(nx) + bn sin(nx)] 2 n=1 (1) verify that the Fourier coefficients an and bn are given by the formulas an = 1 f (x) cos(nx) dx , bn =
- 1 f (x) sin(nx) dx,
- 2. (i) Check that the functions einx , n Z form an orthogonal system in the complex valued functions L2 [-, ]. (ii) Normalize them to obtain an orthonormal system. (iii) Assuming that f L2 [-, ] has its Fourier series f=
n=- ^ fn einx (2) verify that the Fourier coefficients are given by 1 ^ fn = 2 f (x)e-inx dx
- ^ (iv) Verify that the Fourier coefficients an , bn , fn of a function f (x) are related by the formulas ^ ^ ^ ^ an = fn + f-n for n = 0, 1, 2, . . . , bn = i(fn - f-n ) for n = 1, 2, . . . 1 ^ (v) Find the conditions on fn that ensure that the function f (x) is real valued. 4. Use a change of the variable x in (2) to show that the Fourier series of a function g L2 [a, b] has the form g=
n=- gn e2inx/(b-a) ^ and find the formula that expresses gn in terms of g(x). ^ 5. The Legendre orthogonal polynomials are orthogonal in L2 [-1, 1]. Use a Gram-Schmidt process on the polynomials 1, x, x2 , x3 to obtain an orthonormal set; these are the first four Legendre polynomials. 6. The Laguerre orthogonal polynomials are orthogonal in the weighted L2 ([0, +), e-x dx). Use a Gram-Schmidt process on the polynomials 1, x, x2 , x3 to obtain an orthonormal set; these are the first four Laguerre polynomials. 2 ...
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